In this note we introduce the notion of smooth module to extend results from homology theory of Banach algebras to the locally convex category. A complete locally convex module over an m-convex algebra is shown to be smooth if and only if it is topologically isomorphic to a reduced inverse limit of Banach modules over Banach algebras. Stability properties for smoothness are discussed and conditions under which an arbitrary locally convex module is rendered smooth are given.
It is known that the bidual of a quasinormable Fréchet space E with local Banach spaces (En ) n ∈N is topologically isomorphic to the inverse limit of E n n ∈N . With the aid of the Arens product and by homological means, we prove that the previous result is equally valid for quasinormable Fréchet m-convex algebras. This allows showing that the bidual of a σ-C * -algebra equipped with the Arens product is a σ-C * -algebra and presenting a new direct proof of a result on acyclic spectra due to Palamodov.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.